Answer :
To estimate the distance traveled by the car from [tex]\( t = 0 \)[/tex] to [tex]\( t = 60 \)[/tex] seconds using a left-endpoint Riemann sum with six intervals, follow these steps:
1. Identify the Time Intervals and Velocities:
We are given time intervals at every 10 seconds: 0, 10, 20, 30, 40, and 50 seconds. The velocities recorded at these intervals are: 183.9, 168.0, 106.6, 100.8, 124.5, and 176.1 miles per hour.
2. Determine the Width of Each Interval:
The width of each time interval ([tex]\(\Delta t\)[/tex]) is consistently 10 seconds.
3. Apply the Left-Endpoint Riemann Sum:
In a left-endpoint Riemann sum, we use the velocity at the beginning of each interval to estimate the distance for that interval. For each interval, multiply the velocity at the left endpoint by the width of the interval and add up these products.
The distance for each interval is calculated as follows:
- From [tex]\( t = 0 \)[/tex] to [tex]\( t = 10 \)[/tex]: [tex]\( 183.9 \times 10 \)[/tex]
- From [tex]\( t = 10 \)[/tex] to [tex]\( t = 20 \)[/tex]: [tex]\( 168.0 \times 10 \)[/tex]
- From [tex]\( t = 20 \)[/tex] to [tex]\( t = 30 \)[/tex]: [tex]\( 106.6 \times 10 \)[/tex]
- From [tex]\( t = 30 \)[/tex] to [tex]\( t = 40 \)[/tex]: [tex]\( 100.8 \times 10 \)[/tex]
- From [tex]\( t = 40 \)[/tex] to [tex]\( t = 50 \)[/tex]: [tex]\( 124.5 \times 10 \)[/tex]
- From [tex]\( t = 50 \)[/tex] to [tex]\( t = 60 \)[/tex]: [tex]\( 176.1 \times 10 \)[/tex]
4. Sum Up the Distances from Each Interval:
Calculate the total distance by adding all the distances from each interval:
[tex]\[
\text{Total Distance} = (183.9 \times 10) + (168.0 \times 10) + (106.6 \times 10) + (100.8 \times 10) + (124.5 \times 10) + (176.1 \times 10)
\][/tex]
5. Calculate and Round the Final Answer:
Perform the calculations and round the total distance to three decimal places.
The estimated distance traveled by the car from [tex]\( t = 0 \)[/tex] to [tex]\( t = 60 \)[/tex] seconds is [tex]\( 8599.0 \)[/tex].
1. Identify the Time Intervals and Velocities:
We are given time intervals at every 10 seconds: 0, 10, 20, 30, 40, and 50 seconds. The velocities recorded at these intervals are: 183.9, 168.0, 106.6, 100.8, 124.5, and 176.1 miles per hour.
2. Determine the Width of Each Interval:
The width of each time interval ([tex]\(\Delta t\)[/tex]) is consistently 10 seconds.
3. Apply the Left-Endpoint Riemann Sum:
In a left-endpoint Riemann sum, we use the velocity at the beginning of each interval to estimate the distance for that interval. For each interval, multiply the velocity at the left endpoint by the width of the interval and add up these products.
The distance for each interval is calculated as follows:
- From [tex]\( t = 0 \)[/tex] to [tex]\( t = 10 \)[/tex]: [tex]\( 183.9 \times 10 \)[/tex]
- From [tex]\( t = 10 \)[/tex] to [tex]\( t = 20 \)[/tex]: [tex]\( 168.0 \times 10 \)[/tex]
- From [tex]\( t = 20 \)[/tex] to [tex]\( t = 30 \)[/tex]: [tex]\( 106.6 \times 10 \)[/tex]
- From [tex]\( t = 30 \)[/tex] to [tex]\( t = 40 \)[/tex]: [tex]\( 100.8 \times 10 \)[/tex]
- From [tex]\( t = 40 \)[/tex] to [tex]\( t = 50 \)[/tex]: [tex]\( 124.5 \times 10 \)[/tex]
- From [tex]\( t = 50 \)[/tex] to [tex]\( t = 60 \)[/tex]: [tex]\( 176.1 \times 10 \)[/tex]
4. Sum Up the Distances from Each Interval:
Calculate the total distance by adding all the distances from each interval:
[tex]\[
\text{Total Distance} = (183.9 \times 10) + (168.0 \times 10) + (106.6 \times 10) + (100.8 \times 10) + (124.5 \times 10) + (176.1 \times 10)
\][/tex]
5. Calculate and Round the Final Answer:
Perform the calculations and round the total distance to three decimal places.
The estimated distance traveled by the car from [tex]\( t = 0 \)[/tex] to [tex]\( t = 60 \)[/tex] seconds is [tex]\( 8599.0 \)[/tex].
